I can answer that. It all has to do with rival pariarchates.
There were quite a few leading Bishops around mesopotamia in the early Church. Theoretically they were under the control of Antioch once the notion of the Pentarchy developed. In 410 the Christian communities of Mesopotamia renounced all subjection to Antioch and the “Western” bishops and the Bishop of Seleucia-Ctesiphon assumed the rank of Catholicos (later, Patriarch of Babylon was added to that). This was the consolidation of the Assyrian Church of the East. In 431 at the Council of Ephesus, Nestorius was condemned. The Assyrian Church refused to ratify this condemnation, thus becoming the first group to break of from the One Church.
In the 15th century, The Assyrian Church made the Patriarchy hereditary. A group of Assyrians who did not agree with this decision elected a rival patriarch, who sought communion with the Catholic Church in Rome. This was accepted, although he eventually returned to the Assyrian Church. Then, the Pope appointed a new Patriarch to replace him. These people under him, and the other Assyrians who joined under him over time, became the Chaldean Catholic Church.
The second most recent Patriarch of the Assyrian Church of the East, Mar Eshai Shimun XXIII fled to the US during the Assyrian diaspora. In 1973, he became convinced he was allowed to marry based on canon law, and took a wife.
Because of some some liturgical reforms Mar Eshai Shimun made, in 1968 Mar Thoma Darmo, based at Trichur, split from the Church of the East, and became head of the **Ancient Assyrian Church of the East **in 1968 and relocated his see to Baghdad.
Mar Eshai Shimun XXIII tried to resign due to health reasons but was urged to stay. In 1975, he was assassinated. In a council following his death, the hereditary patriarchate was abolished and the current patriarch, Mar Dinkha IV, former bishop of Tehran, to the patriarchate. Mar Dinkha, as Patriarch of theAssyrian CHurch of the East, signed a common Christological document with JPII in the 90s, and this year declared his Church’s intent to unite with the Chaldean Catholic Church, and thus, the Bishop of Rome as well.